Internet hyperbolic mapping paper by Krioukov

Sustaining Internet with Hyperbolic Mapping

- Marian Boguna, Fragkiskos Papadopoulos & Dmitri Krioukov

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Overview

• Hyperbolic Map –basic philosophy• Routing Scheme – Greedy Forwarding• Results-stretch, %shortest path, RT• Model to map current Internet on hyperbolic

Space

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Hyperbolic Map-Basic Philosophy

• Coordinate system in geometric space!• Assign AS coordinates in some geometric space. • Use the space to forward information packets in the

right directions towards their destinations.

• Hyperbolic Map• Angular coordinate- as per geography• Radial coordinate – as a function of the AS degree,

making the space hyperbolic• Only information that ASs must maintain is the

coordinates of their neighbours

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Greedy routing (Kleinberg)

• Greedy forwarding implements routing in the right direction: upon reading the destination address in the packet, the current packet holder forwards the packet to its neighbour closest to the destination in the space

• Uses local neighborhood information only.

Advantages- Greedy Routing

• Greedy routing in complex networks, like the real AS Internet, embedded in hyperbolic spaces, is always successful and always follows shortest paths

• Even if some links are removed, emulating topology dynamics, greedy routing finds remaining paths if they exist.

• No recomputation of node coordinates required.

Results

• Percentage of successful greedy paths 99.99% • Avg Stretch - 1.1• Routing Information at AS AS Degree • approximately the same traffic load on nodes as

shortest-path forwarding• Percentage of successful greedy paths after removal of

X% of links or nodes• X=10% 99%• X=30% 95%

The Model:Einsteinian

• The main property of hyperbolic geometry is the exponential expansion of space expansion of space

• Angular node density

is uniform, but in radial

the number of nodes

grows exponentially

as we move away from

the origin.

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Einsteinian Model

By hyperbolic law of cosines,

Length(ab) is –

coshxab = coshra coshrb-sinhrasinhrbcosab;

ab – is the angle between Oa and Ob.

• rO = 0 in formula gives, Oa = ra and Ob = rb.

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Methods

•  

10/5/06 Lecture #12: Inter-Domain Routing 11

Methods Cont..

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10/5/06 Lecture #12: Inter-Domain Routing 12

Methods Cont..

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10/5/06 Lecture #12: Inter-Domain Routing 13

Internet Topology

• 23572 Ass, 58416 AS links• avg AS degree=4.92, and max degree=2778• avg clustering=0.61• hyperbolic disc radius=27• power law exponent=2.1.

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Mapping AS to countries

• From CAIDA AS ranking project. • Uses 2 methods –• 1) IP based: splits the IP address space

advertised by an AS into small blocks, and then maps each block to a country.

• 2) IP and WHOIS based: reports the country where the AS headquarters are located according to the WHOIS database, for AS’s located in many countries.

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Geographic Routing

• AS angular coordinates equal to their geographic coordinates, the radial coordinate is obtained by Einsteinian model according to the relationship between node degrees and radial positions.

• Then greedy forwarding in the three-dimensional hyperbolic space is performed for Routing.

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Traffic & Congestion considerations

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success ratio& average stretchOn removal of a given fraction of AS nodes

20

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Thank You

Metropolis-Hastings

• Compute current likelihood Lc

• Select a random node• Move it to a new random angular coordinate• Compute new likelihood Ln

• If Ln > Lc, accept the move

• If not, accept it with probability Ln / Lc

• Repeat

ji

aij

aij

ijij xpxpL 1)](1[)(

Sensitivity to missing links

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Mapping the real Internetusing statistical inference methods

• Measure the Internet topology properties• Map them to model parameters• Place nodes at hyperbolic coordinates (r,)

• ’s are uniformly distributed on [0,2]• Apply the Metropolis-Hastings algorithm to find ’s

maximizing the likelihood that Internet is produced by the model